Explain conduction in solids in terms of the movement of free (delocalised) electrons in metallic conductors

2.3.1 Conduction in Solids

Learning Objective

Explain conduction in solids in terms of the movement of free (delocalised) electrons in metallic conductors, and relate this to thermal conduction and the behaviour of gases and liquids.

1. The Free‑Electron (Electron‑Sea) Model

  • In a metal the outer electrons of each atom are not bound to a single nucleus; they become delocalised and form a “sea of electrons” that can move throughout the crystal lattice.

  • The lattice consists of positively charged metal ions arranged in a regular, repeating pattern.

  • Because the electrons are free to roam, they can carry electric charge and also transport kinetic energy (heat).

Typical diagram: a regular array of positive metal ions (blue spheres) with a surrounding cloud of free electrons (red dots). An applied electric field E (arrow) causes the electrons to drift opposite to the field direction.

2. Electrical Conduction – How It Happens

When a potential difference V is applied across a metal, an electric field

\(E = \dfrac{V}{L}\)

is established (L = length of the conductor). Each free electron experiences a force \(-eE\) and acquires a small average drift velocity vd opposite to the field.

Current density:

\(\displaystyle \mathbf{J}=n\,e\,\mathbf{v}_d\)

  • n = number of free electrons per unit volume (≈ 1028 m⁻³ for typical metals)

  • e = elementary charge = 1.60 × 10⁻¹⁹ C

  • vd ≈ 10⁻⁴ m s⁻¹ (very small compared with the random thermal speed)

Macroscopic current for a conductor of cross‑sectional area A:

\(\displaystyle I = J A = n e A v_d\)

3. From the Electron Model to Ohm’s Law

Balancing the electric force with the average resistance due to collisions gives

\(\displaystyle v_d = \frac{e\tau}{m}\,E\)

where m is the electron mass and τ the mean time between collisions.

Substituting \(E = V/L\) leads to a linear relationship between voltage and current:

\(\displaystyle V = I R\)

with

\(\displaystyle R = \rho \frac{L}{A}\qquad\text{and}\qquad

\rho = \frac{m}{n e^{2}\tau}\)

Thus the macroscopic resistance depends on microscopic quantities (n and τ) and on the geometry of the conductor.

4. Factors Influencing Electrical Conductivity

  1. Number of free electrons (n) – More carriers → lower resistivity, higher conductivity.

  2. Mean free time (τ) – Fewer collisions (e.g., at lower temperature) increase τ and reduce ρ.

  3. Cross‑sectional area (A) – Larger area reduces resistance for a given length.

  4. Length (L) – Longer conductors have higher resistance.

  5. Material type – Metals have many free electrons; insulators have essentially none.

5. Thermal Conduction in Solids

5.1 Metals – Free‑Electron (Electron‑Sea) Conduction

  • The same delocalised electrons that carry charge also transport kinetic energy.

  • Electrons move much faster than atoms, so heat is transferred efficiently.

  • This explains why most metals are excellent thermal conductors.

5.2 Non‑metals – Lattice‑Vibration (Phonon) Conduction

  • In ceramics, glasses and polymers the outer electrons remain bound to their atoms.

  • Heat is transferred by collective vibrations of the crystal lattice, called phonons.

  • Phonon transport is slower than electron transport, giving non‑metals lower thermal conductivity (still higher than gases).

Simple sketch of a lattice vibration: neighbouring atoms (grey spheres) oscillate about their equilibrium positions, propagating a vibrational wave (phonon) through the solid.

5.3 Side‑by‑Side Comparison of the Two Mechanisms

AspectMetals (Free‑electron)Non‑metals (Phonon)
Primary carriersDelocalised electronsLattice vibrations (phonons)
Typical thermal conductivity (W m⁻¹ K⁻¹)≈ 200–400 (copper ≈ 400)≈ 0.5–30 (glass ≈ 1)
Dependence on temperatureDecreases with rising temperature (more collisions)Increases with temperature (more phonons)
Effect of impurities/defectsStrongly increase resistivity (electron scattering)Scatter phonons, reducing thermal conductivity

6. Why Gases and Most Liquids Are Poor Electrical Conductors

  • Atoms or molecules are far apart; virtually no free charge carriers exist unless the gas is ionised (plasma) or the liquid contains dissolved ions.

  • Pure liquids such as water have a very low concentration of ions, giving a high resistivity.

  • Result: the number density n of charge carriers is extremely low, so the resistivity is very large.

Real‑world example: High‑voltage transmission lines rely on the insulating properties of air. The large resistivity of the surrounding air prevents arcing between conductors, allowing safe transmission of electricity over long distances.

7. Typical Resistivity and Conductivity Values

MaterialResistivity ρ (Ω·m)Conductivity σ (S·m⁻¹)
Copper1.68 × 10⁻⁸5.96 × 10⁷
Aluminium2.82 × 10⁻⁸3.55 × 10⁷
Silver1.59 × 10⁻⁸6.30 × 10⁷
Iron9.71 × 10⁻⁸1.03 × 10⁷
Glass (insulator)≈ 10¹⁰ – 10¹⁴≈ 10⁻¹⁰ – 10⁻¹⁴

8. Practical Demonstrations (AO3)

8.1 Electrical Conduction – Good vs. Bad Conductors

  • Equipment: two identical rods (copper and wood), a 1.5 V battery, connecting wires, a small bulb, a digital multimeter.

  • Procedure: Connect each rod in turn to the battery–bulb circuit.

  • Observations:

    • Copper rod: bulb glows brightly, measurable current.

    • Wooden rod: bulb remains off, current ≈ 0 A.

  • Explanation: The copper rod contains a high density of free electrons, providing a path for charge flow; the wooden rod does not.

8.2 Thermal Conduction – Good vs. Bad Conductors

  • Equipment: two rods of equal dimensions (metal e.g., aluminium, and plastic), a hot‑plate set to a constant temperature, two thermocouples or digital temperature probes, a stopwatch.

  • Procedure:

    1. Place both rods on the hot‑plate simultaneously.

    2. Record the temperature at the far (upper) end of each rod every 30 s for 5 min.

  • Typical Results (qualitative):

    • Metal rod: rapid temperature rise; reaches near the hot‑plate temperature within a minute.

    • Plastic rod: slow, modest temperature increase; remains considerably cooler.

  • Link to Theory: In the metal the free‑electron sea transports kinetic energy efficiently, whereas in the plastic heat is carried only by phonons, which move more slowly.

9. Comparison with Insulators

  • In insulators the outer electrons remain tightly bound to their atoms → essentially no free charge carriers (n ≈ 0).

  • Resulting resistivity is extremely high; current is negligible under ordinary voltages.

  • Thermal conduction occurs mainly via phonons, which is slower than the electron‑mediated heat transfer in metals.

10. Summary of Key Points

  • Metallic conduction is explained by the free‑electron (electron‑sea) model.

  • Electric current is the drift of these electrons under an applied electric field.

  • Ohm’s law follows from the linear relationship between drift velocity and electric field.

  • Resistivity depends on the number of free electrons and the frequency of their collisions (mean free time τ).

  • Free electrons also carry kinetic energy, giving metals high thermal conductivity; non‑metals rely on lattice‑vibration (phonon) conduction.

  • Gases and most liquids lack free charge carriers, so they are poor electrical conductors; this property is exploited in many practical situations (e.g., air as an insulator in power lines).

11. Practice Questions

  1. A copper wire 2.0 m long has a cross‑sectional area of \(1.0\times10^{-6}\,\text{m}^2\). Calculate its resistance. (Use \(\rho_{\text{Cu}} = 1.68\times10^{-8}\,\Omega\!\cdot\!m\).)

  2. Explain why the resistance of a metal typically decreases when it is cooled.

  3. Compare the conduction mechanisms in a metal and a semiconductor.

  4. Why are gases such as air poor conductors? Give one practical example where this property is useful.

  5. Describe how the same free electrons that conduct electricity also contribute to heat flow in a metal.

  6. In a thermal‑conduction experiment, a metal rod reaches 80 °C after 2 min on a hot‑plate, whereas a plastic rod reaches only 30 °C. Explain the difference using the concepts of free‑electron and phonon conduction.

12. Answers to Practice Questions

  1. \(R = \rho \dfrac{L}{A} = (1.68\times10^{-8})\dfrac{2.0}{1.0\times10^{-6}} = 3.36\times10^{-2}\,\Omega\).

  2. Cooling reduces lattice vibrations, increasing the mean free time τ. From \(\rho = \dfrac{m}{n e^{2}\tau}\), a larger τ lowers resistivity, so the resistance falls.

  3. Metal: free electrons are always present and move under an electric field (electron‑sea).

    Semiconductor: electrons must be excited across a band gap; conduction involves electrons in the conduction band and holes in the valence band, making conductivity strongly temperature‑dependent.

  4. Gases have an extremely low density of free charge carriers; without ionisation they cannot support a continuous current. Example: Air acts as an insulator between high‑voltage transmission lines, preventing arcing.

  5. When an electric field pushes electrons, they also carry kinetic energy from hotter to cooler regions. Because electrons move much faster than lattice vibrations, they transfer heat efficiently, giving metals their high thermal conductivity.

  6. The metal rod conducts heat mainly via its free electrons, which transport energy rapidly, so the temperature rises quickly. The plastic rod relies on phonon (lattice‑vibration) conduction, which is slower, so its temperature increase is modest.